Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 00 Dec 27, 2014.

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1 Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals Gary D. Simpson gsim100887@aol.com rev 00 Dec 27, 2014 Summary Definitions are presented for "quaternion functions" of a quaternion. Polynomial and exponential quaternion functions are presented. Derivatives and integrals of these quaternion functions are developed. It is shown that quaternion multiplication can be represented by matrix multiplication provided the matrix has a specific type of structure. It is also shown that differentiation and integration are similar to their non-quaternion applications. Preface Knowledge of quaternions and Linear Algebra is required. Page 1 of 21

2 Discussion In Part II of "Elements of Quaternions", Hamilton develops the Calculus of quaternions. His thinking in that text seems to have been restricted to situations where the coefficients of a quaternion are functions of an independent parameter such as time. It does not appear that he considered the possibility of a quaternion being a function of another quaternion. It also does not appear that he considered the derivative of a quaternion taken with respect to a quaternion. The objective of this text is to extend Hamilton s work to include these possibilities. Development Background: Let us begin by defining quaternions X and Y such that: and = = + = = + Here, a quaternion is designated by a bold-faced CAPITAL letter. A vector is designated by a bold-faced lower-case letter. A scalar is designated by a lower-case letter in regular font. Quaternions are used to describe physical, three-dimensional space. The unit vectors i, j, and k form a three-dimensional coordinate system. The six values x i, x j, x k, y i, y j, and y k each represent distances in the direction of the respective unit vectors. But what of the values x 0 and y 0? The author speculates that these values are starting points. Therefore, the physical position denoted by quaternion X is the result of beginning at point x 0 and moving distances x i, x j, and x k in the direction of the respective unit vectors. A similar statement is true for quaternion Y. Thought of in this way, the values for x 0 and y 0 default to zero. Now let us suppose that X and Y represent separate coordinate systems. Let us further suppose that the i-j-k orientations of the coordinate systems are equal. Essentially, the systems are the same except they have different starting points (i.e., the coordinate systems are translated). For such a system, the starting points x 0 and y 0 are related as follows: = + = + These relations allow either X or Y to be expressed with respect to the reference point of the other coordinate system. The vector difference associated with the translation is merged with the other vector terms of the respective quaternion. Page 2 of 21

3 Quaternions were developed by Hamilton as a method of expressing the ratio between two vectors. Therefore, let us now consider that application. Let the quaternion Q represent the ratio between vector y and vector x. The quaternion Q is therefore: = = = ; = =0 If both sides are multiplied (on the right-hand side of Q) by the denominator, the result is: Equation 0: = = + + When this multiplication is performed, the result is the following system of equations: + + = 0 + = + = + = The objective is to solve for the four terms of Q. Therefore, this system is expressed as the following matrix multiplication: Equation 0.1: = 0 If the transpose of the coefficient matrix is pre-multiplied by the coefficient matrix, the result is a diagonal matrix as follows: = Therefore, the inverse of the coefficient matrix is simply the transpose matrix divided by the sum of the squares of x i, x j, and x k. This allows the coefficients of quaternion Q to be determined as follows: Page 3 of 21

4 Equation 0.2: = Equation 0.2.1: = Equation 0.2.2: = Equation = + + Equation 0.2.4: = When discussing the interpretation of the scalar term of X, the author speculated that the x 0 term is the starting point and the vector portion of X is added to that point to produce a final position. So then, what is the meaning of Equation since it describes the scalar value q 0 produced by the ratio y/x? The units of measure (or lack thereof) associated with Equations are a clue. The dimensions of the individual x terms and y terms are all length. Therefore, q 0, q i, q j, and q k are each dimensionless since they have units of length 2 /length 2. This should not be a surprise since Q is the ratio between two space vectors and the vectors units of measure (length) should cancel when the ratio is taken. Therefore, the terms of Q do not have a spatial meaning. The meaning of Q is simply to convert between x and y. Pre-multiplication of vector x by quaternion Q maps vector x into vector y. There is something very noteworthy regarding the terms of Q. The sum of the three xy terms in Equation is the dot product of the vector x and the vector y. The other xy terms in Equations form the cross product of the vector x and the vector y. The sum of the squares of the three terms of x is the square of the length of vector x. Therefore, quaternion Q can be expressed as follows: Equation 0.2.5: = = 1 + ; = = Page 4 of 21

5 The subscript R is being used to designate right-hand side multiplication. The author will note that both Newton s Law of Gravity and Coulomb s Law of Electrostatics are inversely proportional to the square of the distance. Equation incorporates this feature with the added bonus of both scalar and vector forms. Therefore, Equation could be used to describe - but not explain - both forces provided the vectors x and y are properly selected and provided consideration is given to the system units of measure and the direction of action. The author will also note that in electro-magnetism, the Lorentz Force is described as F = q(e + v x B) where F is the resulting force, q is the particle charge, E is the electric field, v is the particle velocity, and B is the magnetic field. It seems clear to the author that the quaternion mathematics of Hamilton is a tool that can describe many of the observed physical laws of nature. Next let us consider the case where Q is the ratio between two quaternions. = = = = Repeating the exercise above produces the following system of equations: Equation 0.3: = If the transpose of the coefficient matrix is pre-multiplied by the coefficient matrix, the result is a diagonal matrix as follows: = Therefore, the inverse of the coefficient matrix is simply the transpose matrix divided by the sum of the four squares. This allows the coefficients of quaternion Q to be determined as follows: Equation 0.4: = Page 5 of 21

6 Equation 0.4.1: = Equation 0.4.2: = Equation = Equation 0.4.4: = Therefore, the quaternion Q can be expressed as follows: Equation 0.4.5: = = A consequence of this analysis is that the units of measure associated with the scalar x 0 must be the same as the units of measure associated with the vector x. A similar requirement is placed upon the scalar y 0 and the vector y. In Equation 0, it would have been equally valid to place the quaternion X to the left of Q instead of to the right of Q. Therefore, this exercise will be repeated for that form. Equation 0.1 becomes: Equation 0.5: = + 0 Page 6 of 21

7 Multiplication by the inverse matrix then produces: Equation 0.6: = This then allows the values for the coefficients of Q to be determined as follows: Equation 0.6.1: Equation 0.6.2: Equation 0.6.3: Equation 0.6.4: = = + + = = + + A quick comparison between the right-hand Q and the left-hand Q indicates that they share the same scalar term but that they have opposite vector terms. Therefore, Equation 0.6.5: = = 1 ; = = The subscript L is being used to designate left-hand side multiplication. The quaternions Q R and Q L are complex conjugates. Continuing the left-hand side evaluation for the ratio between two quaternions produces the following: Page 7 of 21

8 Equation 0.7: = This then allows the values for the coefficients of Q to be determined as follows: Equation 0.7.1: Equation 0.7.2: Equation 0.7.3: Equation 0.7.4: = = = = Therefore, the quaternion Q can be expressed as follows: Equation 0.7.5: = = Next, let us define a generic quaternion function F(X) such that Y = F(X). In principle, F(X) could be anything although the author is primarily considering functions such as polynomials and exponentials. The author will not attempt to develop the sine and cosine functions. Matrices will be used extensively. The utility of the matrix formulation will become apparent during the discussion of differentiation and integration. Page 8 of 21

9 Polynomials: In algebra, one of the simplest functions is the line. This is usually presented as y = mx + b. The quaternion equivalent of this is: The multiplication MX is as follows: The detailed steps are as follows: =+ = The next concept is absolutely crucial to understanding all of the material presented here. The quaternion multiplication MX can be represented as a matrix multiplication as follows: = = The quaternion characteristics (i.e., the values 1, i, j, and k) are contained in the column matrix [x]. The square matrix [m] contains only scalar values. Formulating the problem is this manner allows the matrix methods of linear algebra to be combined with quaternions. Now, the quaternion line can be expressed as follows: Equation 1: + =+= = = = + The author requests that the reader carefully examine the 4x4 matrix [m] above. It should be noted that the matrix [m] consists of four 2x2 matrices as follows: Equation 1.1: + = + + Page 9 of 21

10 Where Equation 1.1.1: = ; = This is a special type of symmetry that appears in Linear Algebra when dealing with matrix inversion. Equation 1 can be solved for X as follows: = where [y], [m], [x], and [b] are as presented in Equation 1. This requires the determination of the inverse of matrix [m]. At first glance, this would appear to be difficult, since it requires solving for the 16 terms of the inverse matrix. Fortunately, the symmetry of the problem (as presented in Equation 1.1) simplifies this greatly. Multiplication of the coefficient matrix by its transpose produces a diagonal matrix. For ease of reading, the author will substitute the letters A, B, C, and D for the terms of the coefficient matrix. =? = Therefore, the inverse matrix is simply the transpose matrix divided by the sum of the four squares. Equation 1.2: = = Now let us very briefly consider the complex conjugate of M. The conjugate is usually denoted using the asterisk symbol (*). The complex conjugate of M is therefore denoted as M*. Conjugates are defined as follows: Equation 1.2.1: = + ; = The term m represents the vector portion of M. The conjugate is noteworthy because [m] T = [m*]. For the special case where the transpose matrix is also the inverse matrix, the sum of the four squares must be equal to one. This follows directly from Equation 1.2. Page 10 of 21

11 For quaternions, the order of multiplication is important. Therefore, the reader might reasonably ask what would be the result if the order of multiplication in Equation 1 were to be changed from MX to XM instead? The multiplication XM is as follows: The detailed steps are as follows: = If the terms are rearranged so as to preserve the MX matrix form of Equation 1, these detailed steps become: This multiplication is then expressed as a matrix as follows: This coefficient matrix does not match either the coefficient matrix from Equation 1 or the transpose of that matrix. Instead, this matrix is a curious blend of the two. Column 1 and row 1 are from the original coefficient matrix. The remainder is from the transpose. This makes the author curious regarding the possible forms that the MX multiplication of Equation 1 can take. These are MX, M*X, XM, and XM*. They can be represented by the following matrix multiplications: = Page 11 of 21

12 = = = The author anticipates that these forms will be useful in the future. If the four coefficient matrices are added together, the result is a diagonal matrix with the value 4m 0 along the diagonal. Next, let us consider a simple quadratic such as Y = X 2. This can be thought of as being Y = XX. Therefore, Equation 1 can be used with M = X and B = 0. + = = Now suppose that Y = X 3. This can be written as Y = XX 2. This can then be combined with the relation for X 2 to produce: By logical extension, it follows that: Equation 1.3: + = = = = For N = 1, the coefficient matrix becomes the Identity Matrix. For N = 0, the coefficient matrix becomes the Inverse Matrix. The value of N is restricted to integer values since the (N - 1) term that is used as an exponent on the coefficient matrix must be an integer. The above relations make it possible to produce a generic quaternion-based polynomial as follows: Page 12 of 21

13 Equation 1.4: = where the various terms can be inferred from Equation 1 and Equation 1.3. As an example, consider the following: = = Multiplication is associative for matrices and quaternions. Exponentials: Now let us consider exponentials of a quaternion. We will begin with Euler's Equation. = cos + sin Since cos(-x) = cos(x) and sin(-x) = -sin(x), Euler's Equation can also be written as: = cos sin The i (italicized i) term used in this equation is normally thought to be the complex i with it being defined by i 2 = -1. Part of Hamilton's definitions for the unit vectors i, j, and k is that i 2 = j 2 = k 2 = -1 = ijk. Given the similarity of these definitions, would it be correct to substitute one of Hamilton's unit vectors for the complex i in Euler's Equation? The author will present several pieces of evidence that support the affirmative. The essence of the author s argument is that the behavior of Euler s Equation remains true, but the complex i remains distinct from the unit vectors i, j, and k. The following few paragraphs will support this thinking. Since i 2 = i 2, the most that could be stated would be that i = ±i. If i is substituted into Euler s Equation for i, the ± behavior of the sine and cosine terms considers this, and Euler s Equation remains valid. The same reasoning applies for the unit vectors j and k. For the unit vectors, i 2 = j 2 = k 2 = -1 (= i 2 ). Despite this definition, i j k since the unit vectors are in different directions. The unit vectors are distinct from each other despite being defined by the same equality. There is no reason to think that the complex i is not distinct also, despite being defined by the same equality. Quaternions are composed of four terms with one term being scalar and the other three terms being vectors. It seems reasonable to think that the complex i is associated with the scalar term. This would provide some symmetry between the scalar and vector portions. Page 13 of 21

14 Consider the following: = cos + sin = cos sin Multiplication of Euler s Equation by one of the unit vectors can produce a vector as shown immediately above. The dot product of two vectors is equal to the product of their lengths multiplied by the cosine of the angle between them. The cross product of two vectors is equal to the product of their lengths multiplied by the sine of the angle between them multiplied by a unit vector perpendicular to the vectors. Therefore, it is possible to construct Euler s Equation using the dot product and cross product of two arbitrary vectors as follows: = cos + sin= 1 + ; =0, =0 In each example presented here, Euler s Equation works equally well whether the complex i is used or a unit vector is used. Therefore, the author is convinced that Euler s Equation can be considered to be a quaternion with a single vector term. The next step is to extend Euler s Equation to include the unit vectors j and k. Ideally, the author hopes for a relationship similar to the following: Equation 2: Where: Equation 2.1: = cos +sin +sin +sin = + + Since Euler s Equation does not include a scalar term as part of the exponential, the author will exclude it for now. A scalar term can easily be incorporated in a later step. If Equation 2 and Equation 2.1 are combined, the result is: Equation 2.2: = = = cos +sin +sin +sin Euler s Equation can be applied individually to the unit vectors as follows: Equation 2.3: = cos + sin cos + sin cos + sin Page 14 of 21

15 The terms on the right-hand side of Equation 2 can now be determined by performing the multiplication presented in Equation 2.3. The results of this multiplication are as follows: Equation 2.4.1: Equation 2.4.2: Equation 2.4.3: Equation 2.4.4: cos = cos cos cos sin sin sin sin = sin cos cos + cos sin sin sin = cos sin cos sin cos sin sin = sin sin cos + cos cos sin This system of equations can also be represented by the following matrix multiplication (see Polynomials above): Equation 2.5: cos cos sin 0 0 cos 0 sin 0 cos sin sin = sin = cos cos 0 sin cos sin sin 0 cos 0 sin 0 0 sin cos 0 sin 0 cos 0 sin A generic quaternion exponential is then produced as follows: Equation 2.6: where: Equation 2.6.1: = = = + Therefore, the exponential of q 0 becomes a pre-multiplier for Equation 2.5 as follows: Equation 2.7: cos cos sin 0 0 cos 0 sin 0 cos = sin sin = sin cos cos 0 sin cos sin sin 0 cos 0 sin 0 0 sin cos 0 sin 0 cos 0 sin Page 15 of 21

16 In the discussion of the complex i, the author speculated that the complex i was associated with the scalar term. Equation 2.7 supports this. The complex i could operate upon the q 0 term. Differentiation: In this section the author will discuss and develop a method of differentiation as it applies to the quaternion functions presented above. The author begins with a classical definition followed by development of the differential operator. The author then presents the derivatives of the functions presented above. The first objective is to determine a meaning for the 1st derivative dy/dx. The definition of the 1st derivative is: Equation 3: = lim + The difficulty of this equation is that both X and Y contain four terms. The author will present two methods of determining the derivative. Method 1: Since both the numerator and the denominator of the right-hand side of Equation 3 are quaternions, and since the ratio between two quaternions is also a quaternion, the derivative on the left-hand side must be a quaternion. Therefore, let us make the following definitions: Equation 3.1: = = = = and Equation 3.2: = Multiplying Equation 3.1 by this definition for dx produces: Equation 3.3: = = There is some ambiguity here since the order of multiplication is important for the unit vectors and the author has chosen to place the dx to the right of F'(X). Page 16 of 21

17 Now let us define dy as follows: Equation 3.4: Substituting this into Equation 3.3 produces: Equation 3.5: = = When the multiplication on the right-hand side is performed, the result is a system of four simultaneous equations that can be represented in matrix form as follows: Equation 3.6: = or as Equation 3.7: = ; = Equation 3.7 is a compact way of expressing Equation 3.6. The elements of [f ] are all scalar values. The quaternion attributes are contained in dx and dy. The derivative of Y with respect to X may be thought of as a quaternion or as a matrix. The matrix structure ensures that this is true. Please note the following: The coefficient matrix in Equation 3.6 has the same matrix structure as the coefficient matrix in Equation 1 in the discussion of Polynomials. In multi-variable Calculus, there is a distinction between the total derivative and the various partial derivatives. The total derivative is defined as follows: Equation 3.8: = Therefore, the coefficient matrix presented in Equation 3.6 can be represented by a system of partial derivatives as follows: Page 17 of 21

18 Equation 3.8.1: = Now let us consider again the definition for the derivative presented by Equation 1. = lim + It seems clear that this derivative can be determined by individually incrementing each of the terms of Y by each of the individual terms of X. The resulting matrix is a Jacobian Matrix. It also seems clear that the differential operator d/dx must operate as follows: Equation 3.8.2: = = = or alternately Equation : = = = Page 18 of 21

19 Higher order derivatives are determined by successive substitutions. For example, the 2nd derivative is determined as follows: = = = and = Method 2: This method is based upon the definition presented by Equation 3. = lim + Rather than use the Δ as a difference operator and then take the limit as it reduces to zero, the author proposes to multiply the quaternion X by a scalar value λ and then take the limit as λ goes to zero. This will assure that all of the coefficients of X go to zero together. Equation 3 is therefore re-written as follows: Equation 3.9: =lim + ; Now let us begin the discussion regarding the differentials of quaternion polynomials and quaternion exponentials as described in the sections Polynomials and Exponentials. It will quickly be seen that the Equation 3.9 form of the definition is very simple to use. We will begin with the constant function Y = A. This is the zero quaternion. The next function to consider is Y = AX. =lim =lim = =lim + =lim = Page 19 of 21

20 The next polynomial function is Y = AX 2. =lim + =lim 1+ =lim 2+ =2 The next function to consider is Y = AX 3. =lim + =lim 1+ =lim =3 At this point, it is clear that quaternion differentials of quaternion polynomials behave exactly the same as their non-quaternion counterparts. The next functions to consider are the exponentials. Let Y = e X. = = lim 0 = lim 0 1 = lim 0 1 = lim 0 = The author used L' Hôpital's Rule in the above. It seems that differentiation of a quaternion exponential is also identical to the non-quaternion counterpart. Integration: Given the similar behavior of differentiation for quaternion and non-quaternion functions, integration of a quaternion function is expected to behave like its non-quaternion counterpart. The only variations that the author expects concern the constant of integration and the limits of integration. When indefinite integration is used, a constant of integration must be added to the result. For quaternion integration, this constant of integration must be a quaternion. It is of course possible that this constant quaternion has zero for its scalar and/or vector portions. When definite integration is used, the limits of integration must satisfy the quaternion function. For example, consider the following: = This can easily be solved to give V(T 1 - T 0 ) = (X 1 - X 0 ). The puzzling aspect of this is that there are so very many values for T and X that cannot satisfy this equation. This will be more fully developed in future work. Page 20 of 21

21 The next observation concerns the order of multiplication of the integrand. The usual convention in Calculus would be to write the integrand as V dt rather than dt V. As was seen in the section on Polynomials, reversing the order of multiplication will produce the complex conjugate. Therefore, the complex conjugate is also a solution to the integration. Both solutions will satisfy the equation: = The author anticipates that this can be applied to such concepts as quantum spin and uncertainty. Acknowledgements The author thanks Susan Barlow, Karoly Kehrer, and the rest of the WSM user's group on Yahoo.com. The author also thanks vixra.org and Wikipedia. Page 21 of 21

Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals. Gary D. Simpson. rev 01 Aug 08, 2016.

Quaternion Dynamics, Part 1 Functions, Derivatives, and Integrals Gary D. Simpson gsim1887@aol.com rev 1 Aug 8, 216 Summary Definitions are presented for "quaternion functions" of a quaternion. Polynomial